A guided missile is launched to destroy a fighter jet (Fig. 1).

Fig. 1

We introduce a coordinate axes such that at the missile is at origin and the jet at . The jet flies parallel to the x-axis with constant speed . The missile has locked onto the jet so it is always pointing at the jet as it moves. Its speed is

Find the time and position the missile strikes its target.

“The missile has locked onto the jet so it is always pointing at the jet as it moves” means that the tangent to the missile’s path at any point will pass through the position of the jet. The equation of the tangent is

In addition,

Notice are functions of time

Differentiate (1) with respecte to

By (1), substituting for ,

Let we express (2) as

Suppose , which implies that (see* Exercise-4*). Solving for gives

Submitting (4) into (3),

Let

(5) becomes

We solve this non-linear differential equation as follows:

For ,

Since (See “I vs. CAS“),

(6) gives

At

(7) yields

Moreover, since , we have

Using Omega CAS Explorer:

we obtain

Since , (8) gives

Suppose when the missile hits the target. Then the striking coordinates are the same as that of the fighter jet, i.e.,

Hence,

For (i.e., ), as (8) gives (see *Exercise-1*)

That is,

By (see below),

Since , we have

Namely,

It follows that the guided missile strikes the fighter jet at

(see “Deriving Two Inverse Functions“)

(see “Introducing Lady L” and “Two Peas in a Pod, Part 3“)

i.e.,

*Exercise-1* Show that for , (8) gives (Hint: (10))

*Exercise-*2 For , what is the total distance traveled by the missile when it strikes the fighter jet? (hint: Don’t make things harder than they are)

*Exercise-3* Show that if (i.e., ), the missile will not strike the fighter jet. Explain.

*Excercise-4* Show that .

*Excercise-5* For , find the time and position the missile strikes its target .